\[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (A -\frac {a \left (b^{2} B -a b C +a^{2} D\right )}{b^{3}}\right ) x^{5}}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (2 A \,b^{3}+a \left (5 b^{2} B -12 a b C +19 a^{2} D\right )\right ) x^{5}}{35 a^{2} b^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {a \left (b C -3 a D\right ) x}{3 b^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\left (2 b C -9 a D\right ) \arctanh \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{2 b^{\frac {11}{2}}}-\frac {\left (4 b C -15 a D\right ) x}{3 b^{5} \sqrt {b \,x^{2}+a}}+\frac {D x \sqrt {b \,x^{2}+a}}{2 b^{5}} \]
command
integrate(x**4*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________